Interdependent transport via percolation backbones in spatial networks
IInterdependent transport via percolation backbones in spatial networks
Bnaya Gross, ∗ Ivan Bonamassa, and Shlomo Havlin Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel (Dated: September 7, 2020)The functionality of nodes in a network is often described by the structural feature of belongingto the giant component. However, when dealing with problems like transport, a more appropriatefunctionality criterion is for a node to belong to the network’s backbone, where the flow ofinformation and of other physical quantities (such as current) occurs. Here we study percolationin a model of interdependent resistor networks and show the effect of spatiality on their coupledfunctioning. We do this on a realistic model of spatial networks, featuring a Poisson distribution oflink-lengths. We find that interdependent resistor networks are significantly more vulnerable thantheir percolation-based counterparts, featuring first-order phase transitions at link-lengths wherethe mutual giant component still emerges continuously. We explain this apparent contradictionby tracing the origin of the increased vulnerability of interdependent transport to the crucialrole played by the dandling ends. Moreover, we interpret these differences by considering anheterogeneous k -core percolation process which enables to define a one-parameter family offunctionality criteria whose constraints become more and more stringent. Our results highlightthe importance that different definitions of nodes functionality have on the collective properties ofcoupled processes, and provide better understanding of the problem of interdependent transport inmany real-world networks. This work is dedicated to the late Prof. Dietrich Stauffer from whom we learned a lot aboutpercolation.
Keywords: Resistor networks, Interdependent networks, Percolation theory, Spatial networks
I. INTRODUCTION
Throughout the last decades, network science has pro-vided important tools to study complex systems such asthe brain [1, 2], climate networks [3, 4], protein interac-tions [5, 6] and finance [7–9], offering a powerful frame-work for exploring their collective phenomena [10]. Theability to simplify a complex system to its basic ingredi-ents and still observing the general phenomenon occur-ring in it is, perhaps, one of the main reasons for the riseof network science in recent years.A prominent tool commonly used in the analysis ofthe structure and function of many real-world networksis percolation theory [11–13]. During this process, a frac-tion 1 − p of nodes or edges are randomly removed andcertain quantities of interest such as the giant compo-nent (GC), the correlation length, or the susceptibility,are then measured. For sufficiently large values of p , agiant component spanning the entire network exists, en-abling the communication between nodes belonging to it,and at a critical threshold p c it dismantles into a collec-tion of small clusters.The functionality of the network is usually describedby adopting as a proxy the relative size of the GC, P ∞ , and nodes that disconnect from it are isolated andconsidered as non-functional. However, when transportprocesses like e.g. current flow in resistor networks [14–16] are considered, a more appropriate criterion for the ∗ [email protected] 𝑉 𝑉 AB FIG. 1:
Illustration of the interdependent resistormodel . Transport of currents in two networks of spatial re-sistors, A and B, are mutually coupled via dependency links(dashed lines). Each layer is constructed with links of thesame characteristic length ζ as described in Eq. (1) and thesame average degree z , though their local wiring features aregenerally different. The backbone in each layer consists ofthe red nodes connected via the red connectivity links whichconduct current between the network’s boundaries. The bluenodes do not conduct current (dead ends) and thus belongsonly to the giant component but not to the backbone. A nodewill fail if it is not part of the backbone of its network or ifits dependent node in the other layer fails. nodes’ functionality has to be introduced. For resistornetworks, such condition can be identified in the require-ment that a node belongs to the relative size of thenetwork’s backbone [12, 16], B ∞ , which contains onlyconduct-current nodes, i.e. no dead-ends (see Fig. 1).The importance of these differences in the definition ofthe nodes’ functionality becomes more significant whenconsidering multilayer networks [17, 18] and, in partic-ular, interdependent networks [19–24]. In such cases,the failing of a node in one network can cause furtherdamage in other one, which can in its turn trigger a cas- a r X i v : . [ phy s i c s . s o c - ph ] S e p cade of failures resulting in abrupt collapses signaled byfirst-order structural transitions. Since failed nodes arethe ones spreading the damage from one network to theother, the precise definition of the functionality criterionbecomes a crucial ingredient in understanding the vul-nerability and characterizing the functional regimes ofinterdependent systems.In this paper, we study percolation on a model oftwo interdependent resistor networks with conductivity-based states, so that global transport is attained onlyif a mutual backbone exists. Motivated by recent evi-dence on transport networks [25, 26] and in the connec-tome’s structure of mammals [27–30], we consider herethe realistic case of spatially embedded networks with atunable characteristic link length [25, 31–33]. We findthat, in contrast to a single network where both the GCand backbone have the same critical threshold, in inter-dependent networks the critical thresholds signalling thecollapse of the giant components are different. In par-ticular, we show that the critical threshold for the back-bone is much higher compared with its percolation-basedanalogue, hinting at the extreme vulnerability of inter-dependent transport in spatially embedded networks. Inaddition, while the transition changes from second to firstorder as the interaction range (link-length) increases forboth the GC and backbone, the backbone transition be-comes first order in a much shorter interaction range com-pared to the GC. Furthermore, using heterogeneous k-core percolation [34–36], we are able to explain the reasonfor the shorter interaction range required to trigger first-order transitions in interdependent resistor networks. Weshow that as the criteria for node functionality gets morestrict, the damage can spread in the whole system withshorter interaction range.We stress that the cost function here considered ismore realistic than those of previous studies [37], andmotivated by data-driven evidence reported in transportsystems [25, 26] and in brain networks [27–30]. In this re-spect, our results provide additional insights to the prop-erties of coupled transport processes in spatial infrastruc-tures [38–40], offering a simple and realistic framework toinvestigate their robustness. II. THE MODEL
We model interdependent transport by means of twospatial networks, A and B, as depicted in Fig. 1. Thenodes in each layer are placed on a 2-dimensional grid ofsize N = L × L , where L is the grid length, on the posi-tions ( x, y ) where x, y ∈ [0 , L −
1] are integers numbers.The connectivity links in each network are then assignedby picking randomly a node i and connecting it with arandom node j at Euclidean distance d ij drawn from anexponential distribution P ij ( d ij ) ∝ exp( − d ij /ζ ) . (1) FIG. 2:
Percolation and conductivity thresholds in asingle spatial network.
Both P ∞ and B ∞ have the samepercolation threshold p c for any value of ζ . Adopting z = 4,one has in the limit of ζ (cid:28) p Dc (cid:39) . ζ → ∞ any pair of nodes can be con-nected with the same probability similar to an ER networkwith p ERc = 1 /z . The inset shows the size of the giant com-ponent (GC) and the backbone of a single 2D lattice. Noticethat the GC contains also nodes that do not conduct current(dead ends, see Fig. 1) and thus the backbone is a sub-set ofthe GC, while the transition occurs at the same percolationthreshold. Here and throughout the paper, simulation resultsare obtained for networks of size N = 10 . Here, ζ represents the characteristic link-length of thenetwork and plays the role of a tunable parameter con-trolling the influence of spatiality on the range of interac-tions. This picking process repeats until a given averagedegree z is reached. As discussed in earlier works bysome of us [33, 41], the structure of the network signifi-cantly depends on the characteristic link-length ζ : whilesmall values of ζ produce strongly space-dependent net-works, large values of ζ (order O ( L )) produce networkswith weak space-dependence which can be analyzed viamean-field approaches [25, 31–33]. The two networks de-pend on each other through dependency links betweennodes placed in the same geometrical position in bothnetworks (see Fig. 1). Therefore, if the node ( x, y ) failsin layer A, then also its “replicated” node ( x, y ) in layerB will fail. Let us stress that the neighbourhoods of su-perposed nodes in the two layers are generally different,since each layer is a different instance of the same statis-tical ensemble of spatially embedded networks.We study percolation on our interdependent modelwith conductivity-based functionality by removing non-conducting nodes that do not belong to the percola-tion backbone (the dandling ends) of each layer, whichwe measure by searching for the networks’ largest bi-components [42]. The process is initiated by removinga fraction 1 − p of nodes from network A. This removalmay disconnect some nodes from the backbone of net-work A causing their dependent nodes in network B tobe removed as well. The removal of nodes in network Bmay disconnect more nodes from the backbone of net-work B which, in their turn, make their dependent nodesin network A to fail, hence propagating the damage. Thisrepeating cascade of failures describes the dynamic be-havior of the system and it is an intrinsic property ofinterdependent networks and their stability. Once thecascading process stops, the remaining active nodes inthe whole system form the mutual backbone (MB). Simi-larly, the remaining active nodes after the cascading pro-cess with only percolation-based functionality form theso-called mutual giant component (MGC) of the system.Notice that, although both the MB and the MGC are re-spectively, subsets of the backbone and giant componentin their isolated counterparts, they are measured respec-tively by means of the very same observables, namely B ∞ and P ∞ . III. THE EFFECTS OF FUNCTIONALITY ONTHE PHASE TRANSITION
To understand the significant difference betweenpercolation-based functionality and conductivity-basedfunctionality in interdependent networks, let us first con-sider the case of a single isolated layer. Percolation in asingle network yields a continuous structural transitionat the same position for both the GC and the backbone(see Fig. 2). The reason is that a path from one sideof the network to the other exists even if non-conductingnodes (dandling ends) are removed [11, 12] and thus theirremoval only affect the magnitude of the order parame-ter without changing the transition threshold (see Fig. 2,inset). In the limit of ζ (cid:28)
1, only short link-lengths areallowed and a 2D lattice-like structure is created, with p Dc (cid:39) . ζ → ∞ , any pair of nodes canbe connected with the same probability similar to an ERnetwork, leading therefore to the percolation threshold p ERc = 1 /z . Notice that p c rapidly converges towards p ERc (see Fig. 2), resulting in a 2D-to-random crossoverwith surprising features, whose details were extensivelyaddressed in Ref. [33].Interdependent networks experience completely differ-ent phenomena compared to a single network. For thecase of the percolation-based functionality [25], in thelimit ζ → ∞ (two interdependent ER networks) the per-colation phase transition becomes first-order as shownin Fig. 3a, and it can be analytically solved, resultingin the critical threshold p c (cid:39) . /z [19] (see Fig. 4).Moreover, a tricritical characteristic length ζ c (cid:39)
12 ex-ists above which a local damage will propagate at dis-tances sufficiently large (i.e. larger than the radius of acritical droplet [43–45]) igniting a percolative nucleationprocess [25] that leads to a first-order phase transition.In contrast, for ζ < ζ c local failures generally remainconfined, leading to continuous phase transitions whosecluster statistics is strongly influenced by finite-size ef- FIG. 3:
Interdependent percolation and conductivitytransitions.
The relative size of the (a) MGC, P ∞ , andthe (b) MB, B ∞ , as a function of p for several values of ζ are shown. For small values of ζ the transition is continuousfor both the MGC and the MB. However, as ζ exceeds acritical interaction length, ζ c , the transition becomes first-order. Notice that ζ c of the MGC is larger compared to thatof the MB. fects [46].The case of conductivity-based interdependence, dis-closes important differences compared to its percolation-based analogue. The first difference can be identified inthe transition point, which is not in the same position ascan be seen in Fig. 3. This is in marked contrast to a sin-gle network case where the transition point is in the sameposition (Fig. 2). The reason for this difference can beunderstood in the effect of the dangling ends. For a singlenetwork, joining the network’s boundaries exists even af-ter the removal of the dangling ends, thus, their removaldoes not affect the transition threshold. However, oncedependency links between networks are set, the removalof the dandling ends in one network can lead to failureof nodes belonging to the backbone of the other network,a genuine multilayer effect that finds no analogy in theisolated case. This removal leads to a much strongercascade of failures in the system compare to percolation-based functionality and breaks the path joining the net-work’s boundaries. These cascades lead to the separationof the transition of percolation-based functionality andconductivity-based functionality observed even at smallinteraction ranges and it further explains the origin un-derlying the extreme vulnerability of the MB. Anotherimportant difference between the MGC and the MB isthe transition behaviour for different values of ζ . Simi-larly to the MGC, the MB undergoes a first-order tran-sition for ζ → ∞ as shown in Fig. 3b whose featurescan be solved analytically (see Appendix), resulting inthe threshold p c (cid:39) . /z as shown in Fig. 4. How-ever, the value of ζ c is much smaller: while for the MGC, ζ c (cid:39)
12, for the MB, ζ c (cid:39)
6, as shown in Fig. 4.
IV. TRICRITICAL POINTS IN THECHARACTERISTIC RANGE OFINTERACTIONS
In order to better understand the drastic decrease ofthe tricritical interaction range ζ c for conductivity-based FIG. 4:
Critical thresholds.
Phase diagram showing thecritical thresholds p c for the MGC (red) and the MB (blue)change with increasing values of ζ . For ζ < ζ c both transi-tions are continuous and p c increases close to linearly with ζ , reaching a peak at ζ c . For ζ > ζ c , the transitions are in-stead first-order and both p c slowly decrease, converging tothe mean-field value. In the limit of random interactions, i.e. ζ → ∞ , p c → . /z for the MGC (red dashed line) and p c ≈ . /z (blue dashed line) for the MB for z = 4. No-tice that the value of ζ c of the MB is smaller compared tothat of the MGC (approximately 6 for MB and 12 for MGC),unveiling a region where, even if the MGC undergoes a contin-uous phase transition, the MB collapses abruptly. The insetdemonstrates this phenomenon for ζ = 8. functionality systems, we here examine heterogeneous k -core percolation on our interdependent spatial networkmodel. Let us recall that k -core percolation is an itera-tive process initiated by random removal of 1 − p fractionof nodes followed by iterative removal of nodes with de-gree less than k until only the k -core remains [47]. In heterogeneous k -core percolation, the degree threshold isnot the same for all the nodes [34–36], and it is assigned ina way such that an r fraction of randomly chosen nodeshas threshold k a and the remaining fraction 1 − r hasthreshold k a + 1. Thus, the average degree threshold isgiven by k = k a (1 − r ) + r ( k a + 1) . (2)By continuously increasing r , we study the effect of nodefunctionality on the system’s phase transitions as it getsincreasingly more stringent. Eq. (2), in fact, allows toidentify a one-parameter family of functionality criteriafor each r so that different levels of functionality con-strains can be compared.We start with the case, k a = 1 and r = 0, which cor-responds to the MGC, and increase r to study how thetricritical interaction range ζ c will change with k . Thephase diagram in Fig. 5 discloses the dependence of the FIG. 5:
Phase diagram for interdependent k -core. p c is measured as a function of ζ for different values of averagedegree threshold, k , as calculated from Eq. (2) with k a = 1.As expected for k = 1 the case of MGC is recovered with ζ c (cid:39)
12. However, as k increases and the nodes functionalitycriterion gets more strict, ζ c decreases as shown by the blackline. The case of k = 2 shows the same critical interactionrange as the MB ( ζ c ≈
6) even though 2-core percolation andthe backbone are not exactly the same since node can havedegree 2 but not be part of the backbone. percolation threshold as a function of ζ for different aver-age degree thresholds. As expected, for r = 0 and k = 1we find ζ c (cid:39)
12, as in Fig. 4 for the MGC. However, as theaverage degree threshold, k , increases, ζ c decreases. Thisshow that as the node functionality gets more strict, notonly that the percolation threshold increases but the crit-ical interaction range decreases. In other words, the MBhas a much lower tricritical interaction range comparedto the MGC , leading to a cascade of failures and abruptcollapses already at a relatively small range of interac-tions. In line with evidence raised by previous results ininterdependent transport processes in spatial networks[38–40], our results highlight the dramatic fragility of in-frastructures and transport systems. V. SUMMARY AND DISCUSSION
In this work, we have studied the effect of spatialityon interdependent resistor networks emphasizing the dif-ferences between percolation-base functionality governedby the GC and conductivity-based functionality governedby the backbone. Our model makes a step forward to-wards a more realistic characterization of interdependenttransport processes in real-world systems, thanks to therealistic spatial topology we have considered. We findthat while in a single network the percolation transi-tion is the same for both functionality criterion, oncedependency links are formed between networks the tran-sition thresholds are significantly different with highervulnerability for the backbone. Moreover, both crite-ria have a tricritical interaction length above which thestructural transitions are first-order and continuous be-low. We also find that the tricritical interaction lengthfor the MB is shorter compared to that of the MGC,highlighting the extreme vulnerability of interdependenttransport processes [38–40]. We have explained this dif-ference by adopting a model of interdependent hetero-geneous k -core, showing that the tricritical interactionrange decrease as the criterion for the nodes functional-ity gets more strict.Our results highlight the crucial role played by the defi-nition of node functionality which significantly affects itsrobustness against random failures, and offer new per-spectives regarding the influence that precise definitionsof nodes’ functionality can have on their coupled collec-tive phenomena. For example, a system of real interde-pendent networks might be characterized by percolation-based functionality in one layer and conductivity-basedfunctionality in another, an outcome that would lead tocritical features in between the two cases studied here.Moreover, in systems with even stricter node functional-ity criteria, e.g. governed by heterogeneous k -core with k > VI. ACKNOWLEDGEMENTS
We thank the Israel Science Foundation, the BinationalIsrael-China Science Foundation Grant no. 3132/19,ONR, the BIU Center for Research in Applied Cryptog-raphy and Cyber Security, NSF-BSF Grant no. 2019740,and DTRA Grant no. HDTRA-1-19-1-0016 for financialsupport.
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